This determines the maximum number of simultaneous requests that can be handled. If not specified, this attribute is set to the default value of 200.

How the process works:

At server startup, the HTTP Connector will create a number of processing threads based on the value configured for the minSpareThreads attribute.

Each incoming request requires a thread for the duration of that request.

If the number of simultaneous requests cannot be handled by the currently available request processing threads, additional threads will be created up to the configured maximum (the value of the maxThreads attribute).

If still more simultaneous requests are received, they are stacked up up to the configured maximum (the value of the acceptCount attribute).

Any further simultaneous requests will receive "connection refused" errors, until resources are available to process them.

Guidelines for maxThreads:maxThreads is an important tuning parameter, however if you are reaching an error like:

you should at first investigate if it's rather a problem of individual requests taking too long: are your threads returning to the pool? if, for example, database connections are not released, threads pile up waiting to obtain a database connection thereby making it impossible to process additional requests. This is a problem in your webapp. Take a thread dump to find out where they're stuck.
Increasing too much maxThreads will lead to :

Consume a good chunk of memory.

Your system will spend too much time context switching

So once you have already optimized your application try raising you maxThread attribute up to 500-750. I wouldn't advice to create larger Connectors, rather if 750 Connections are not enough create a Cluster configuration with several Tomcat instances. For example 2 instances of tomcat each one with maxThreads=500 instead of a single Tomcat with maxThreads=1000

Monitoring Tomcat maxThreads

Once logged into the Manager application (See this tutorial to learn how to create un user for the Manager application), click on the Server Status link. At the bottom of the Server status link you will see this information:

There you can gather important information such as:

The number of Max threads

The current thread count

The current thread busy

The Max processing time

The Request count

The Error count

The Bytes received

The Bytes sent

If you want to learn more about JDK tuning read this tutorial which contains many tips valid also for Tomcat:JBoss Performance Tuning